AudienceThis is an introductory textbook dealing with the design and analysis of experiments. It is basedon college-level courses in design of experiments that I have taught over nearly 40 years atArizona State University, the University of Washington, and the Georgia Institute of Technology.It also reﬂects the methods that I have found useful in my own professional practice as an engineering and statistical consultant in many areas of science and engineering, including the researchand development activities required for successful technology commercialization and productrealization.The book is intended for students who have completed a ﬁrst course in statistical methods. This background course should include at least some techniques of descriptive statistics,the standard sampling distributions, and an introduction to basic concepts of conﬁdenceintervals and hypothesis testing for means and variances. Chapters 10, 11, and 12 requiresome familiarity with matrix algebra.Because the prerequisites are relatively modest, this book can be used in a second courseon statistics focusing on statistical design of experiments for undergraduate students in engineering, the physical and chemical sciences, statistics, mathematics, and other ﬁelds of science.For many years I have taught a course from the book at the ﬁrst-year graduate level in engineering. Students in this course come from all of the ﬁelds of engineering, materials science,physics, chemistry, mathematics, operations research life sciences, and statistics. I have alsoused this book as the basis of an industrial short course on design of experiments for practicing technical professionals with a wide variety of backgrounds. There are numerous examplesillustrating all of the design and analysis techniques. These examples are based on real-worldapplications of experimental design and are drawn from many different ﬁelds of engineering andthe sciences. This adds a strong applications flavor to an academic course for engineersand scientists and makes the book useful as a reference tool for experimenters in a varietyof disciplines.

v

vi

Preface

About the BookThe eighth edition is a major revision of the book. I have tried to maintain the balancebetween design and analysis topics of previous editions; however, there are many new topicsand examples, and I have reorganized much of the material. There is much more emphasis onthe computer in this edition.

Design-Expert, JMP, and Minitab SoftwareDuring the last few years a number of excellent software products to assist experimenters inboth the design and analysis phases of this subject have appeared. I have included output fromthree of these products, Design-Expert, JMP, and Minitab at many points in the text. Minitaband JMP are widely available general-purpose statistical software packages that have gooddata analysis capabilities and that handles the analysis of experiments with both ﬁxed and random factors (including the mixed model). Design-Expert is a package focused exclusively onexperimental design. All three of these packages have many capabilities for construction andevaluation of designs and extensive analysis features. Student versions of Design-Expert andJMP are available as a packaging option with this book, and their use is highly recommended. I urge all instructors who use this book to incorporate computer software into yourcourse. (In my course, I bring a laptop computer and use a computer projector in everylecture, and every design or analysis topic discussed in class is illustrated with the computer.)To request this book with the student version of JMP or Design-Expert included, contactyour local Wiley representative. You can ﬁnd your local Wiley representative by going towww.wiley.com/college and clicking on the tab for “Who’s My Rep?”

Empirical ModelI have continued to focus on the connection between the experiment and the model thatthe experimenter can develop from the results of the experiment. Engineers (and physical,chemical and life scientists to a large extent) learn about physical mechanisms and theirunderlying mechanistic models early in their academic training, and throughout much oftheir professional careers they are involved with manipulation of these models.Statistically designed experiments offer the engineer a valid basis for developing anempirical model of the system being investigated. This empirical model can then bemanipulated (perhaps through a response surface or contour plot, or perhaps mathematically) just as any other engineering model. I have discovered through many years of teachingthat this viewpoint is very effective in creating enthusiasm in the engineering communityfor statistically designed experiments. Therefore, the notion of an underlying empiricalmodel for the experiment and response surfaces appears early in the book and receivesmuch more emphasis.

Factorial DesignsI have expanded the material on factorial and fractional factorial designs (Chapters 5 – 9) inan effort to make the material flow more effectively from both the reader’s and the instructor’s viewpoint and to place more emphasis on the empirical model. There is new materialon a number of important topics, including follow-up experimentation following a fractionalfactorial, nonregular and nonorthogonal designs, and small, efficient resolution IV and Vdesigns. Nonregular fractions as alternatives to traditional minimum aberration fractions in16 runs and analysis methods for these design are discussed and illustrated.

Preface

vii

Additional Important ChangesI have added a lot of material on optimal designs and their application. The chapter on responsesurfaces (Chapter 11) has several new topics and problems. I have expanded Chapter 12 onrobust parameter design and process robustness experiments. Chapters 13 and 14 discussexperiments involving random effects and some applications of these concepts to nested andsplit-plot designs. The residual maximum likelihood method is now widely available in software and I have emphasized this technique throughout the book. Because there is expandingindustrial interest in nested and split-plot designs, Chapters 13 and 14 have several new topics.Chapter 15 is an overview of important design and analysis topics: nonnormality of theresponse, the Box – Cox method for selecting the form of a transformation, and other alternatives; unbalanced factorial experiments; the analysis of covariance, including covariates in afactorial design, and repeated measures. I have also added new examples and problems fromvarious ﬁelds, including biochemistry and biotechnology.

Experimental DesignThroughout the book I have stressed the importance of experimental design as a tool for engineers and scientists to use for product design and development as well as process development and improvement. The use of experimental design in developing products that are robustto environmental factors and other sources of variability is illustrated. I believe that the use ofexperimental design early in the product cycle can substantially reduce development lead timeand cost, leading to processes and products that perform better in the ﬁeld and have higherreliability than those developed using other approaches.The book contains more material than can be covered comfortably in one course, and Ihope that instructors will be able to either vary the content of each course offering or discusssome topics in greater depth, depending on class interest. There are problem sets at the endof each chapter. These problems vary in scope from computational exercises, designed toreinforce the fundamentals, to extensions or elaboration of basic principles.

Course SuggestionsMy own course focuses extensively on factorial and fractional factorial designs. Consequently,I usually cover Chapter 1, Chapter 2 (very quickly), most of Chapter 3, Chapter 4 (excludingthe material on incomplete blocks and only mentioning Latin squares brieﬂy), and I discussChapters 5 through 8 on factorials and two-level factorial and fractional factorial designs indetail. To conclude the course, I introduce response surface methodology (Chapter 11) and givean overview of random effects models (Chapter 13) and nested and split-plot designs (Chapter14). I always require the students to complete a term project that involves designing, conducting, and presenting the results of a statistically designed experiment. I require them to do thisin teams because this is the way that much industrial experimentation is conducted. They mustpresent the results of this project, both orally and in written form.

The Supplemental Text MaterialFor the eighth edition I have prepared supplemental text material for each chapter of the book.Often, this supplemental material elaborates on topics that could not be discussed in greater detailin the book. I have also presented some subjects that do not appear directly in the book, but anintroduction to them could prove useful to some students and professional practitioners. Some ofthis material is at a higher mathematical level than the text. I realize that instructors use this book

viii

Preface

with a wide array of audiences, and some more advanced design courses could possibly beneﬁtfrom including several of the supplemental text material topics. This material is in electronic formon the World Wide Website for this book, located at www.wiley.com/college/montgomery.

WebsiteCurrent supporting material for instructors and students is available at the websitewww.wiley.com/college/montgomery. This site will be used to communicate informationabout innovations and recommendations for effectively using this text. The supplemental textmaterial described above is available at the site, along with electronic versions of data setsused for examples and homework problems, a course syllabus, and some representative student term projects from the course at Arizona State University.

Student Companion SiteThe student’s section of the textbook website contains the following:1. The supplemental text material described above2. Data sets from the book examples and homework problems, in electronic form3. Sample Student Projects

Solutions to the text problemsThe supplemental text material described abovePowerPoint lecture slidesFigures from the text in electronic format, for easy inclusion in lecture slidesData sets from the book examples and homework problems, in electronic formSample SyllabusSample Student Projects

The instructor’s section is for instructor use only, and is password-protected. Visit theInstructor Companion Site portion of the website, located at www.wiley.com/college/montgomery, to register for a password.

Student Solutions ManualThe purpose of the Student Solutions Manual is to provide the student with an in-depth understanding of how to apply the concepts presented in the textbook. Along with detailed instructions on how to solve the selected chapter exercises, insights from practical applications arealso shared.Solutions have been provided for problems selected by the author of the text.Occasionally a group of “continued exercises” is presented and provides the student with afull solution for a speciﬁc data set. Problems that are included in the Student SolutionsManual are indicated by an icon appearing in the text margin next to the problem statement.This is an excellent study aid that many text users will ﬁnd extremely helpful. TheStudent Solutions Manual may be ordered in a set with the text, or purchased separately.Contact your local Wiley representative to request the set for your bookstore, or purchase theStudent Solutions Manual from the Wiley website.

A Regression ModelComparisons Among Treatment MeansGraphical Comparisons of MeansContrastsOrthogonal ContrastsScheffé’s Method for Comparing All ContrastsComparing Pairs of Treatment MeansComparing Treatment Means with a Control

An ExampleStatistical Analysis of the Fixed Effects ModelModel Adequacy CheckingEstimating the Model ParametersChoice of Sample SizeThe Assumption of No Interaction in a Two-Factor ModelOne Observation per Cell

The General Factorial DesignFitting Response Curves and SurfacesBlocking in a Factorial DesignProblems

IntroductionThe 22 DesignThe 23 DesignThe General 2k DesignA Single Replicate of the 2k DesignAdditional Examples of Unreplicated 2k Design2k Designs are Optimal DesignsThe Addition of Center Points to the 2k DesignWhy We Work with Coded Design VariablesProblems

Confounding the 2k Factorial Design in Two BlocksAnother Illustration of Why Blocking Is ImportantConfounding the 2k Factorial Design in Four BlocksConfounding the 2k Factorial Design in 2p BlocksPartial ConfoundingProblems

8

Two-Level Fractional Factorial Designs8.18.2

8.38.4

8.58.6

320

IntroductionThe One-Half Fraction of the 2k Design

320321

8.2.18.2.28.2.3

321323324

Deﬁnitions and Basic PrinciplesDesign ResolutionConstruction and Analysis of the One-Half Fraction

The One-Quarter Fraction of the 2k DesignThe General 2kϪp Fractional Factorial Design

SUPPLEMENTAL MATERIAL FOR CHAPTER 1S1.1 More about Planning ExperimentsS1.2 Blank Guide Sheets to Assist in Pre-ExperimentalPlanningS1.3 Montgomery’s Theorems on Designed Experiments

The supplemental material is on the textbook website www.wiley.com/college/montgomery.

1.1

Strategy of ExperimentationObserving a system or process while it is in operation is an important part of the learningprocess, and is an integral part of understanding and learning about how systems andprocesses work. The great New York Yankees catcher Yogi Berra said that “. . . you canobserve a lot just by watching.” However, to understand what happens to a process whenyou change certain input factors, you have to do more than just watch—you actually haveto change the factors. This means that to really understand cause-and-effect relationships ina system you must deliberately change the input variables to the system and observe thechanges in the system output that these changes to the inputs produce. In other words, youneed to conduct experiments on the system. Observations on a system or process can leadto theories or hypotheses about what makes the system work, but experiments of the typedescribed above are required to demonstrate that these theories are correct.Investigators perform experiments in virtually all ﬁelds of inquiry, usually to discoversomething about a particular process or system. Each experimental run is a test. More formally,we can deﬁne an experiment as a test or series of runs in which purposeful changes are madeto the input variables of a process or system so that we may observe and identify the reasonsfor changes that may be observed in the output response. We may want to determine whichinput variables are responsible for the observed changes in the response, develop a modelrelating the response to the important input variables and to use this model for process or systemimprovement or other decision-making.This book is about planning and conducting experiments and about analyzing theresulting data so that valid and objective conclusions are obtained. Our focus is on experiments in engineering and science. Experimentation plays an important role in technology

1

2

Chapter 1 ■ Introduction

commercialization and product realization activities, which consist of new product designand formulation, manufacturing process development, and process improvement. The objective in many cases may be to develop a robust process, that is, a process affected minimallyby external sources of variability. There are also many applications of designed experimentsin a nonmanufacturing or non-product-development setting, such as marketing, service operations, and general business operations.As an example of an experiment, suppose that a metallurgical engineer is interested instudying the effect of two different hardening processes, oil quenching and saltwaterquenching, on an aluminum alloy. Here the objective of the experimenter (the engineer) isto determine which quenching solution produces the maximum hardness for this particularalloy. The engineer decides to subject a number of alloy specimens or test coupons to eachquenching medium and measure the hardness of the specimens after quenching. The average hardness of the specimens treated in each quenching solution will be used to determinewhich solution is best.As we consider this simple experiment, a number of important questions come to mind:1. Are these two solutions the only quenching media of potential interest?2. Are there any other factors that might affect hardness that should be investigated orcontrolled in this experiment (such as, the temperature of the quenching media)?3. How many coupons of alloy should be tested in each quenching solution?4. How should the test coupons be assigned to the quenching solutions, and in whatorder should the data be collected?5. What method of data analysis should be used?6. What difference in average observed hardness between the two quenching mediawill be considered important?All of these questions, and perhaps many others, will have to be answered satisfactorilybefore the experiment is performed.Experimentation is a vital part of the scientiﬁc (or engineering) method. Now there arecertainly situations where the scientiﬁc phenomena are so well understood that useful resultsincluding mathematical models can be developed directly by applying these well-understoodprinciples. The models of such phenomena that follow directly from the physical mechanismare usually called mechanistic models. A simple example is the familiar equation for currentﬂow in an electrical circuit, Ohm’s law, E ϭ IR. However, most problems in science and engineering require observation of the system at work and experimentation to elucidate information about why and how it works. Well-designed experiments can often lead to a model ofsystem performance; such experimentally determined models are called empirical models.Throughout this book, we will present techniques for turning the results of a designed experiment into an empirical model of the system under study. These empirical models can bemanipulated by a scientist or an engineer just as a mechanistic model can.A well-designed experiment is important because the results and conclusions that canbe drawn from the experiment depend to a large extent on the manner in which the data werecollected. To illustrate this point, suppose that the metallurgical engineer in the above experiment used specimens from one heat in the oil quench and specimens from a second heat inthe saltwater quench. Now, when the mean hardness is compared, the engineer is unable tosay how much of the observed difference is the result of the quenching media and how muchis the result of inherent differences between the heats.1 Thus, the method of data collectionhas adversely affected the conclusions that can be drawn from the experiment.1

A specialist in experimental design would say that the effect of quenching media and heat were confounded; that is, the effects ofthese two factors cannot be separated.

1.1 Strategy of Experimentation

FIGURE 1.1process or system

Controllable factorsx1

Inputs

x2

z2

General model of a

Outputy

Process

z1

■

xp

3

zq

Uncontrollable factors

In general, experiments are used to study the performance of processes and systems.The process or system can be represented by the model shown in Figure 1.1. We can usuallyvisualize the process as a combination of operations, machines, methods, people, and otherresources that transforms some input (often a material) into an output that has one or moreobservable response variables. Some of the process variables and material properties x1,x2, . . . , xp are controllable, whereas other variables z1, z2, . . . , zq are uncontrollable(although they may be controllable for purposes of a test). The objectives of the experimentmay include the following:1. Determining which variables are most inﬂuential on the response y2. Determining where to set the inﬂuential x’s so that y is almost always near thedesired nominal value3. Determining where to set the inﬂuential x’s so that variability in y is small4. Determining where to set the inﬂuential x’s so that the effects of the uncontrollablevariables z1, z2, . . . , zq are minimized.As you can see from the foregoing discussion, experiments often involve several factors.Usually, an objective of the experimenter is to determine the inﬂuence that these factors haveon the output response of the system. The general approach to planning and conducting theexperiment is called the strategy of experimentation. An experimenter can use several strategies. We will illustrate some of these with a very simple example.I really like to play golf. Unfortunately, I do not enjoy practicing, so I am always looking for a simpler solution to lowering my score. Some of the factors that I think may be important, or that may inﬂuence my golf score, are as follows:1.2.3.4.5.6.7.8.

The type of driver used (oversized or regular sized)The type of ball used (balata or three piece)Walking and carrying the golf clubs or riding in a golf cartDrinking water or drinking “something else” while playingPlaying in the morning or playing in the afternoonPlaying when it is cool or playing when it is hotThe type of golf shoe spike worn (metal or soft)Playing on a windy day or playing on a calm day.

Obviously, many other factors could be considered, but let’s assume that these are the ones of primary interest. Furthermore, based on long experience with the game, I decide that factors 5through 8 can be ignored; that is, these factors are not important because their effects are so small

Chapter 1 ■ Introduction

R

ODriver■

FIGURE 1.2

T

BBall

Score

Score

Score

that they have no practical value. Engineers, scientists, and business analysts, often must makethese types of decisions about some of the factors they are considering in real experiments.Now, let’s consider how factors 1 through 4 could be experimentally tested to determinetheir effect on my golf score. Suppose that a maximum of eight rounds of golf can be playedover the course of the experiment. One approach would be to select an arbitrary combinationof these factors, test them, and see what happens. For example, suppose the oversized driver,balata ball, golf cart, and water combination is selected, and the resulting score is 87. Duringthe round, however, I noticed several wayward shots with the big driver (long is not alwaysgood in golf), and, as a result, I decide to play another round with the regular-sized driver,holding the other factors at the same levels used previously. This approach could be continued almost indeﬁnitely, switching the levels of one or two (or perhaps several) factors for thenext test, based on the outcome of the current test. This strategy of experimentation, whichwe call the best-guess approach, is frequently used in practice by engineers and scientists. Itoften works reasonably well, too, because the experimenters often have a great deal of technical or theoretical knowledge of the system they are studying, as well as considerable practical experience. The best-guess approach has at least two disadvantages. First, suppose theinitial best-guess does not produce the desired results. Now the experimenter has to takeanother guess at the correct combination of factor levels. This could continue for a long time,without any guarantee of success. Second, suppose the initial best-guess produces an acceptable result. Now the experimenter is tempted to stop testing, although there is no guaranteethat the best solution has been found.Another strategy of experimentation that is used extensively in practice is the onefactor-at-a-time (OFAT) approach. The OFAT method consists of selecting a starting point,or baseline set of levels, for each factor, and then successively varying each factor over itsrange with the other factors held constant at the baseline level. After all tests are performed,a series of graphs are usually constructed showing how the response variable is affected byvarying each factor with all other factors held constant. Figure 1.2 shows a set of these graphsfor the golf experiment, using the oversized driver, balata ball, walking, and drinking waterlevels of the four factors as the baseline. The interpretation of this graph is straightforward;for example, because the slope of the mode of travel curve is negative, we would concludethat riding improves the score. Using these one-factor-at-a-time graphs, we would select theoptimal combination to be the regular-sized driver, riding, and drinking water. The type ofgolf ball seems unimportant.The major disadvantage of the OFAT strategy is that it fails to consider any possibleinteraction between the factors. An interaction is the failure of one factor to produce the sameeffect on the response at different levels of another factor. Figure 1.3 shows an interactionbetween the type of driver and the beverage factors for the golf experiment. Notice that if I usethe regular-sized driver, the type of beverage consumed has virtually no effect on the score, butif I use the oversized driver, much better results are obtained by drinking water instead of beer.Interactions between factors are very common, and if they occur, the one-factor-at-a-time strategy will usually produce poor results. Many people do not recognize this, and, consequently,

OFAT experiments are run frequently in practice. (Some individuals actually think that thisstrategy is related to the scientiﬁc method or that it is a “sound” engineering principle.) Onefactor-at-a-time experiments are always less efﬁcient than other methods based on a statisticalapproach to design. We will discuss this in more detail in Chapter 5.The correct approach to dealing with several factors is to conduct a factorial experiment. This is an experimental strategy in which factors are varied together, instead of oneat a time. The factorial experimental design concept is extremely important, and severalchapters in this book are devoted to presenting basic factorial experiments and a number ofuseful variations and special cases.To illustrate how a factorial experiment is conducted, consider the golf experiment andsuppose that only two factors, type of driver and type of ball, are of interest. Figure 1.4 showsa two-factor factorial experiment for studying the joint effects of these two factors on my golfscore. Notice that this factorial experiment has both factors at two levels and that all possiblecombinations of the two factors across their levels are used in the design. Geometrically, thefour runs form the corners of a square. This particular type of factorial experiment is called a22 factorial design (two factors, each at two levels). Because I can reasonably expect to playeight rounds of golf to investigate these factors, a reasonable plan would be to play tworounds of golf at each combination of factor levels shown in Figure 1.4. An experimentaldesigner would say that we have replicated the design twice. This experimental design wouldenable the experimenter to investigate the individual effects of each factor (or the maineffects) and to determine whether the factors interact.Figure 1.5a shows the results of performing the factorial experiment in Figure 1.4. Thescores from each round of golf played at the four test combinations are shown at the cornersof the square. Notice that there are four rounds of golf that provide information about usingthe regular-sized driver and four rounds that provide information about using the oversizeddriver. By ﬁnding the average difference in the scores on the right- and left-hand sides of thesquare (as in Figure 1.5b), we have a measure of the effect of switching from the oversizeddriver to the regular-sized driver, or92 ϩ 94 ϩ 93 ϩ 91 88 ϩ 91 ϩ 88 ϩ 90Ϫ44ϭ 3.25

Driver effect ϭ

That is, on average, switching from the oversized to the regular-sized driver increases thescore by 3.25 strokes per round. Similarly, the average difference in the four scores at the top

Chapter 1 ■ Introduction

88, 91

92, 94

88, 90

93, 91

Type of ball

T

B

O

RType of driver

(a) Scores from the golf experiment+Type of ball–

B

+

+

–

–

T

B

–

+

B

+

+

T

Type of ball

–TType of ball

6

–

ORType of driver

ORType of driver

ORType of driver

(b) Comparison of scores leadingto the driver effect

(c) Comparison of scoresleading to the ball effect

(d) Comparison of scoresleading to the ball–driverinteraction effect

FIGURE 1.5factor effects

■

Scores from the golf experiment in Figure 1.4 and calculation of the

of the square and the four scores at the bottom measures the effect of the type of ball used(see Figure 1.5c):88 ϩ 91 ϩ 92 ϩ 94 88 ϩ 90 ϩ 93 ϩ 91Ϫ44ϭ 0.75

Ball effect ϭ

Finally, a measure of the interaction effect between the type of ball and the type of driver canbe obtained by subtracting the average scores on the left-to-right diagonal in the square fromthe average scores on the right-to-left diagonal (see Figure 1.5d), resulting in92 ϩ 94 ϩ 88 ϩ 90 88 ϩ 91 ϩ 93 ϩ 91Ϫ44ϭ 0.25

Ball–driver interaction effect ϭ

The results of this factorial experiment indicate that driver effect is larger than either theball effect or the interaction. Statistical testing could be used to determine whether any ofthese effects differ from zero. In fact, it turns out that there is reasonably strong statistical evidence that the driver effect differs from zero and the other two effects do not. Therefore, thisexperiment indicates that I should always play with the oversized driver.One very important feature of the factorial experiment is evident from this simpleexample; namely, factorials make the most efﬁcient use of the experimental data. Notice thatthis experiment included eight observations, and all eight observations are used to calculatethe driver, ball, and interaction effects. No other strategy of experimentation makes such anefﬁcient use of the data. This is an important and useful feature of factorials.We can extend the factorial experiment concept to three factors. Suppose that I wishto study the effects of type of driver, type of ball, and the type of beverage consumed on mygolf score. Assuming that all three factors have two levels, a factorial design can be set up

as shown in Figure 1.6. Notice that there are eight test combinations of these three factorsacross the two levels of each and that these eight trials can be represented geometrically asthe corners of a cube. This is an example of a 23 factorial design. Because I only want toplay eight rounds of golf, this experiment would require that one round be played at eachcombination of factors represented by the eight corners of the cube in Figure 1.6. However,if we compare this to the two-factor factorial in Figure 1.4, the 23 factorial design would provide the same information about the factor effects. For example, there are four tests in bothdesigns that provide information about the regular-sized driver and four tests that provideinformation about the oversized driver, assuming that each run in the two-factor design inFigure 1.4 is replicated twice.Figure 1.7 illustrates how all four factors—driver, ball, beverage, and mode of travel(walking or riding)—could be investigated in a 24 factorial design. As in any factorial design,all possible combinations of the levels of the factors are used. Because all four factors are attwo levels, this experimental design can still be represented geometrically as a cube (actuallya hypercube).Generally, if there are k factors, each at two levels, the factorial design would require 2kruns. For example, the experiment in Figure 1.7 requires 16 runs. Clearly, as the number offactors of interest increases, the number of runs required increases rapidly; for instance, a10-factor experiment with all factors at two levels would require 1024 runs. This quicklybecomes infeasible from a time and resource viewpoint. In the golf experiment, I can onlyplay eight rounds of golf, so even the experiment in Figure 1.7 is too large.Fortunately, if there are four to ﬁve or more factors, it is usually unnecessary to run allpossible combinations of factor levels. A fractional factorial experiment is a variation of thebasic factorial design in which only a subset of the runs is used. Figure 1.8 shows a fractionalfactorial design for the four-factor version of the golf experiment. This design requires only8 runs instead of the original 16 and would be called a one-half fraction. If I can play onlyeight rounds of golf, this is an excellent design in which to study all four factors. It will providegood information about the main effects of the four factors as well as some information abouthow these factors interact.

Fractional factorial designs are used extensively in industrial research and development,and for process improvement. These designs will be discussed in Chapters 8 and 9.

1.2

Some Typical Applications of Experimental DesignExperimental design methods have found broad application in many disciplines. As notedpreviously, we may view experimentation as part of the scientiﬁc process and as one of theways by which we learn about how systems or processes work. Generally, we learn througha series of activities in which we make conjectures about a process, perform experiments togenerate data from the process, and then use the information from the experiment to establishnew conjectures, which lead to new experiments, and so on.Experimental design is a critically important tool in the scientiﬁc and engineeringworld for improving the product realization process. Critical components of these activitiesare in new manufacturing process design and development, and process management. Theapplication of experimental design techniques early in process development can result in1.2.3.4.

Experimental design methods are also of fundamental importance in engineeringdesign activities, where new products are developed and existing ones improved. Some applications of experimental design in engineering design include1. Evaluation and comparison of basic design conﬁgurations2. Evaluation of material alternatives3. Selection of design parameters so that the product will work well under a wide variety of ﬁeld conditions, that is, so that the product is robust4. Determination of key product design parameters that impact product performance5. Formulation of new products.The use of experimental design in product realization can result in products that are easierto manufacture and that have enhanced field performance and reliability, lower productcost, and shorter product design and development time. Designed experiments also haveextensive applications in marketing, market research, transactional and service operations,and general business operations. We now present several examples that illustrate some ofthese ideas.

1.2 Some Typical Applications of Experimental Design

EXAMPLE 1.1

Characterizing a Process

A ﬂow solder machine is used in the manufacturing processfor printed circuit boards. The machine cleans the boards ina ﬂux, preheats the boards, and then moves them along aconveyor through a wave of molten solder. This solderprocess makes the electrical and mechanical connectionsfor the leaded components on the board.The process currently operates around the 1 percent defective level. That is, about 1 percent of the solder joints on aboard are defective and require manual retouching. However,because the average printed circuit board contains over 2000solder joints, even a 1 percent defective level results in far toomany solder joints requiring rework. The process engineerresponsible for this area would like to use a designed experiment to determine which machine parameters are inﬂuentialin the occurrence of solder defects and which adjustmentsshould be made to those variables to reduce solder defects.The ﬂow solder machine has several variables that canbe controlled. They include1.2.3.4.5.6.7.

In addition to these controllable factors, several other factorscannot be easily controlled during routine manufacturing,although they could be controlled for the purposes of a test.They are1. Thickness of the printed circuit board2. Types of components used on the board

EXAMPLE 1.2

3. Layout of the components on the board4. Operator5. Production rate.In this situation, engineers are interested in characterizing the ﬂow solder machine; that is, they want to determine which factors (both controllable and uncontrollable)affect the occurrence of defects on the printed circuitboards. To accomplish this, they can design an experimentthat will enable them to estimate the magnitude and direction of the factor effects; that is, how much does theresponse variable (defects per unit) change when each factor is changed, and does changing the factors togetherproduce different results than are obtained from individualfactor adjustments—that is, do the factors interact?Sometimes we call an experiment such as this a screeningexperiment. Typically, screening or characterization experiments involve using fractional factorial designs, such as inthe golf example in Figure 1.8.The information from this screening or characterizationexperiment will be used to identify the critical process factors and to determine the direction of adjustment for thesefactors to reduce further the number of defects per unit. Theexperiment may also provide information about which factors should be more carefully controlled during routine manufacturing to prevent high defect levels and erratic processperformance. Thus, one result of the experiment could be theapplication of techniques such as control charts to one ormore process variables (such as solder temperature), inaddition to control charts on process output. Over time, if theprocess is improved enough, it may be possible to base mostof the process control plan on controlling process input variables instead of control charting the output.

Optimizing a Process

In a characterization experiment, we are usually interestedin determining which process variables affect the response.A logical next step is to optimize, that is, to determine theregion in the important factors that leads to the best possible response. For example, if the response is yield, wewould look for a region of maximum yield, whereas if theresponse is variability in a critical product dimension, wewould seek a region of minimum variability.Suppose that we are interested in improving the yieldof a chemical process. We know from the results of a characterization experiment that the two most importantprocess variables that influence the yield are operatingtemperature and reaction time. The process currently runs

at 145°F and 2.1 hours of reaction time, producing yieldsof around 80 percent. Figure 1.9 shows a view of thetime–temperature region from above. In this graph, thelines of constant yield are connected to form responsecontours, and we have shown the contour lines for yieldsof 60, 70, 80, 90, and 95 percent. These contours are projections on the time–temperature region of cross sectionsof the yield surface corresponding to the aforementionedpercent yields. This surface is sometimes called aresponse surface. The true response surface in Figure 1.9is unknown to the process personnel, so experimentalmethods will be required to optimize the yield withrespect to time and temperature.